The change of variable method is correct.
Put [itex]u = \sqrt t[/itex]
[itex]2du = \frac{dt}{\sqrt t}[/itex]
Substituting this will simplify the integrand. However, I think the mistake you are making is; you are forgetting to change the limits(1 & 1/4) of the integral. Whenever you make a variable change, the limits change according to the new variable.